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“There is a difference between whether the universe is infinite or just really really really really really really big,” Anthony Aguirre said at the recent FQXi conference in Puerto Rico. I’m pretty sure I counted six reallys. With that remark, he encapsulated a major debate going on within physics and cosmology right now. Although the conference theme was officially the physics of information, it could just as well have been the physics of infinity, so often did that little sideways ‘8’ put in an appearance. Is the universe finite or infinite? Is nature capable of a finite or infinite number of possible states? Can spacetime be infinitely subdivided or is it made of finite-size cells? The questions seem undecidable. But maybe the finitude or infinitude makes itself felt every time you do a measurement and every time you stir cream into a coffee cup and can’t unstir it out.

The forces of finitude include Max Tegmark, who has been bad-mouthing infinity on Edge.org, in quotes to New Scientist, and in Chapter 11 of his new book. His complaint is what cosmologists call the measure problem: there’s no way of unambiguously counting members of an infinite set. If there’s no way to count, there’s no way to calculate probabilities and therefore no way to relate theory to experiment. The whole empirical framework of science verges on collapse. A finite universe presents no such difficulty. Even Peter Woit, who agrees with Tegmark on little else, finds common ground with him on the measure problem.

The aficionados of infinity include Alan Guth, who argued in Puerto Rico and on Edge.org that a truly infinite universe would neatly explain the arrow of time. When space has no bound, neither does entropy. It keeps on increasing forever, always pointing the way forward for time. The universe need not have begun in a contrived initial state to create the impetus toward increasing disorder.

When two opposing positions can both muster plausible arguments, what you have is less a debate and more a dilemma. If it were up to them, physicists would surely prefer finitude, yet nature seems to have made different plans. The universe is expanding at an accelerating rate and, if it keeps doing so, it is destined to spawn an infinity of baby universes. “It would be cozy if it were finite, but it doesn’t seem to be,” Aguirre told me. “Eternal inflation gives you an infinite universe, and something like eternal inflation is happening now and probably happened in the past. Nature is rubbing infinity in our face.”

Eternal inflation could cease to be eternal if the dark energy that drives it withered away on a timescale of billions of years. But if dark energy were so unstable, Aguirre has argued, we should see signs of its decay somewhere out there. All indications are that time will never end, which means that space probably doesn’t, either.

The brain-melting Boltzmann-brain paradox is one reason that infinite space and infinite time go together. If time were infinite yet space finite, the contents of the universe would cycle through their possible configurations over and over and over again. Molecules would occasionally converge to produce a conscious mind that lasted for a split second, but was under the misimpression it was the product of billions of years of cosmic evolution. Indeed, in the vastness of eternity, such flashes of deluded awareness would vastly outnumber brains that had formed the old-fashioned way, and we’d have to conclude that our observations are implanted memories, like fossils that young-Earth creationists think God planted in rock strata to fool us. It’s a paradox because an empirical science would lead us to the conclusion that empirical science is a sham.

This sort of argument is what Aguirre had in mind as a genuine distinction between a truly infinite universe and a merely ginormous one. “If it’s finite, no matter how big you make it, it still eventually runs into the paradox,” he said. Chance fluctuations that are inevitable in finite space are vanishingly unlikely in infinite space. An infinite universe is ever-changing, never doing the same thing twice, as Sean Carroll eloquently described in his prize-winning essay for the first FQXi essay contest.

The arrow-of-time argument that Guth has been developing also has the potential of distinguishing infinite from finite. The basic idea goes back to a provocative paper a decade ago by Carroll and Jennifer Chen (who has since left physics research to work on energy regulation). Whereas their scenario involved an accelerating universe, Guth gave a supersimple example involving a gas in an infinite void. At some moment you can take as t=0, the gas occupies some minimum volume. From then on, the gas will expand without limit. If the void is finite, the gas will eventually cycle back to its starting point. Time has a clear forward progression only if the void is truly infinite.

The overall history of Guth’s minimalist universe is fully time-symmetric, as the laws of physics demand. Prior to t=0, the gas was also expanding without limit, albeit backwards in time, and again time has a clear forward progression, the reverse of the arrow on the other side. Only around t=0 does the arrow become ambiguous. If any mortal beings are alive for the crossover, they’ll observe curious reversals of fortune such as those that Ken Wharton, who writes science fiction when not doing physics, once imagined in a poignant short story.

Guth’s scenario is classical, but similar intimations of infinity arise in quantum physics. Yasunori Nomura has argued that an infinite range of possible states (that is, an infinite Hilbert space) would make the process of quantum decoherence irreversible, explaining the arrow of time in quantum measurement.

In a funny way, then, the arrow of time we observe in daily life may reflect the infinity of space, and human mortality may hinge on the immortality of the universe. But still. Infinity? Can it be a real thing rather than simply our idealization?

Aguirre, for all his advocacy of infinity, is unconvinced by Guth’s and Carroll and Chen’s arrow-of-time arguments. “They’re brushing certain things under the rug,” he said. For instance, they take for granted that, if the maximum possible entropy is infinite, it doesn’t matter how the universe began. Any possible initial state has finite entropy, so you get the arrow of time for free. But you can’t take anything for granted when it comes to infinity. Guth and the others implicitly rule out initial states with infinite entropy. Is that really justified? Such a state is hard to imagine, but that doesn’t mean it can’t exist, Aguirre said. His musings remind me of one of the strangest concepts in mathematics: the axiom of choice. This is a rule for selecting objects from an infinite collection even when all standard rules fail. The weird thing is that, although mathematicians know that such a rule exists, they don’t know what the rule is. Worse, they know they’ll never know what it is. A state of infinite entropy may likewise exist even if it is impossible to specify.

Other speakers in Puerto Rico proposed ways to evade the paradoxes that imply infinity. Carroll himself argued that Boltzmann brains go away when you take care to distinguish quantum from thermal fluctuations. Andy Albrecht contended that the measure problem evaporates when you think of all probabilities as inherently quantum. Even a tossed coin, he said, ultimately lands on one side or the other because of quantum indeterminism. If so, probabilities aren’t defined in terms of repeated trials, and the inability to count elements of an infinite set is a red herring.

At this rate, physicists may not have to wonder about infinity. Their discussions may go on long enough to prove the point one way or the other.

" ... there's no way of unambiguously counting members of an infinite set."

My seven year old granddaughter told her mother recently, "Mom, if numbers just go on forever and ever, without stopping, then all numbers are small numbers."

Exactly right. Which is why Leibniz wrote that deep understanding of nature has to be based in the infinitely small. Modern day followers of Leibniz -- Hermann Weyl and Gregory Chaitin among others -- recognize this continuity of form in constructed objects, the continuum in Weyl's purely mathematical terms, the Omega number in Chaitin's mathematical-computational terms.

This is only one more reason why I simply cannot understand why Joy Christian has gotten such a harsh reception, highly undeserved and irrational IMO. His construction which depends on topological non-vanishing torsion is not only one more expression of the infinitely small, it prescribes the exact physical boundary.

" ... there's no way to calculate probabilities and therefore no way to relate theory to experiment."

There's no probability measure in Christian's framework. Like Einstein's mathematically complete theories, it makes a closed logical judgement on the correspondence between mathematical construction and physical result.

The question of finite or infinite universe is not a new one. Cosmologists of today would do better if they studied a bit of history of their own subject. See, for example, "From the Closed World to the Infinite Universe", by Alexandre Koyre.

The sociology of the physics community of today, on the other hand, is a largely new phenomenon. The harsh treatment of my work that you puzzle about can be traced back to the reaction of a tiny number of uninformed postdocs back in 2007 while I was based at the Perimeter Institute. That set a negative tone on my work from which it has yet to recover. The mounting evidence accumulated in its favour has mattered little. That is sociology of science for you. What is accepted and what is rejected in science depends largely on who has a louder voice and a greater political muscle.

Incidentally, Giordano Bruno, whose views on the "plurality of the worlds" are discussed in Koyre's book, did not fair well back in 1600 for his unconventional views either.

"What is accepted and what is rejected in science depends largely on who has a louder voice and a greater political muscle."

Peter Hayes "The Ideology of Relativity: The Case of the Clock Paradox" : Social Epistemology, Volume 23, Issue 1 January 2009, pages 57-78: "The gatekeepers of professional physics in the universities and research institutes are disinclined to support or employ anyone who raises problems over the elementary inconsistencies of relativity. A winnowing out process has made it very difficult for critics of Einstein to achieve or maintain professional status. Relativists are then able to use the argument of authority to discredit these critics. Were relativists to admit that Einstein may have made a series of elementary logical errors, they would be faced with the embarrassing question of why this had not been noticed earlier. Under these circumstances the marginalisation of antirelativists, unjustified on scientific grounds, is eminently justifiable on grounds of realpolitik. Supporters of relativity theory have protected both the theory and their own reputations by shutting their opponents out of professional discourse."

Even if I could excuse them in the past, for not having the mathematics toolbox to "get it," how may one excuse them now -- after all the loud claims that if your framework cannot survive computer simulation, it is worthless?

First, by holding onto probability measures like a monkey caught in a coconut trap.

Second, by failing to acknowledge that Python -- the most general of programming languages -- is capable of treating all objects independently of domain and scale, physical or mathematical.

Third, as illustrated in your spacetime diagram of t --> oo orthogonal to S --> S^3 -- failing to acknowledge the limit of recurrence on the Euclidean manifold, of correlated scale independent pairs of point events on parallelized S^3.

Fourth, and related to the first -- failing to understand that randomly singular events (+/- 1) affect left and right planar trajectories to the extent that no complete result is probabilistic; a coordinate free system of continuous measurement functions is self limiting by the topological constraint of a single point at infinity compactifying R^3.

Fifth, and related to the third and fourth -- failing to understand that the spacetime continuity of relativity obviates faux theories of completeness (as that of our friend who claims to drive nails into the coffin of local deterministic theories), because time is not invariant between inertial frames, as your spacetime diagram illustrates clearly, if one understands what "parallelized 3 sphere" means. The time scalar is never nonlocal, and therefore never invariant among frames of reference, though a physical event at one instant of time will not be affected by a measurement at a later instant of time in the same reference frame.

None of this is beyond the understanding of anyone with elementary knowledge of analysis and topology.

I obtained, many years ago, studying the convergence of a statistical ensemble that follows the gradient back propagation, a system that have an infinite negative entropy; a system that converge to a single state, like a Dirac delta function, have an infinite negative entropy.

If the initial state of the Universe is near a Dirac delta function, then the entropy grow ever: a fluctuation of great complexity, that contain all the story of the Universe, can it possible (we see it); the initial state contain our current consciousness, so that the Boltzmann-brain and this initial fluctuation are equivalent.

I think Max is onto a winner with challenging infinity. Tori aside*, I'm somewhat surprised at the non-sequitur wherein a "flat" universe is assumed to be infinite. Or a nearly-flat universe is assumed to be very large. I think it's a failure of imagination myself, because I've read a lot of the original Einstein material including his Leyden Address, and it's clear that he considered space to be a something rather than a nothing. Take a look at the stress-energy-momentum tensor and there's shear stress and pressure. One can liken space to some kind of gin-clear ghostly elastic. Waves run through it. It expands. The raisin-in-the-cake analogy is like a stress ball when you open your fist. And yet we continue to see the balloon analogy where we've dropped a dimension because we cannot conceive that space might have an edge. Like a droplet of water has an edge, wherein waves suffer total internal reflection. There is no water beyond that edge. There is no space beyond the edge of space.

Since Max's essay was an Edge essay, please can we see some discussion about the edge of the universe?

"...This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials guv), has, I think, finally disposed of the view that space is physically empty..."

I think it's worth explaining the distinction. We talk of curved spacetime in the context of gravitational field, but space isn't curved in the room you're in. It's inhomogeneous. Imagine you've placed an array of light-clocks in an equatorial slice through and around the Earth. When you plot all the clock rates, your plot resembles the "rubber-sheet" depiction you can see on the wiki Riemann curvature tensor article. You measured those clock rates, there's a curvature in your metric. But this "curved spacetime" isn't curved space. It's a curvature in your plot of the inhomogeneity of space. Your lower clocks don't tick slower because your plot of clock rates is curved. They tick slower because the space down here isn't the same as the space up there, because a concentration of energy tied up as the matter of the Earth "conditions" the surrounding space, the effect diminishing with distance.

Can you provide us a link to the Einstein quote? You must be aware that it is claimed that Einstein is the proponent of a relational space which is nothing, in disagreement with the substantivalist view that space is something.

No. If this were true, we would speak of curved time. The curvature of spacetime is the result of a sign change in the metric signature (+ + + - or - - - +) which results from adding the time coordinate (dt)^2 to the set of spatial coordinates which in relativity is x^2 + y^2 + z^2 - (dt)^2

"The quote from Einstein "...This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials guv), has, I think, finally disposed of the view that space is physically empty..."

refers to the ten points of the tensor metric which when time is included in the matrix ("its physical relation")represents a 4-dimensional continuum of spacetime (Minkowski space).

This representation is the genesis of supersymmetric string theory -- for when Theodor Kaluza and Felix Klein formulated a fifth spatial dimension which adds five more elements to the tensor metric for a total of 15 -- we see electromagnetic energy represented as a vibration of the fifth dimension, potentially unifying the theory of gravity with the theory of electromagnetism. String theorists over 30 or 40 years expanded the idea to 10 or 11 spatial dimensions, explaining all physics in a unified model of continuous spacetime as vibrations of higher dimensional reality.

String theory is essentially a theory of nonlocal hidden variables, however, which limits its potential for experimental falsification. Joy Christian provides a measurement framework of local hidden variables in higher dimension topology, which is complete and falsifiable.

Tom: spacetime is a static mathematical model. There is no motion in it because it includes the time dimension. You can draw worldlines in it to represent motion through space over time, but things do not move through spacetime. When Einstein said space, he meant space.

Is it possible that spacetime does only model those relationships? If it is physically unnecessary for it to have/be an actual physical 'fabric,' wouldn't that be conceptually preferable, than insisting this conceptual tool be anything more than that?

Not to keep digging up a sore subject, but it obviously was a useful conceptual tool to model the motions of the heavens as a giant clockworks. That didn't have to mean there was a physical manifestation of this concept, even though it was a popular assumption at the time.

" Relativity, however, is a kinetic theory"

So why dismiss the underlaying kinetics, just because modeling it requires conceptual stasis?

A book is a useful narrative tool, but that doesn't mean the events continue to exist in some parallel universe.

Could you, please, elaborate a bit on Kaluza-Klein 5th dimension? Is it an arbitrary construct, or does it have a relational argument of pertaining to some observed or empirically derived phenomenon? Thanks, jrc

" ... please, elaborate a bit on Kaluza-Klein 5th dimension? Is it an arbitrary construct, or does it have a relational argument of pertaining to some observed or empirically derived phenomenon?"

It isn't arbitrary. It's an extension of the spacetime field. One cannot actually measure events in other than 3 dimensions; however, the time-distance coordinate is a way of describing an object's place in time as well as space, so the precise location of an object in spacetime requires 4 dimension coordinates -- x, y, z, t.

The fifth spatial dimension described by K-K tells us that the effects of electromagnetic energy can be explained by a combination of Einstein's gravity field equations and Maxwell's EM field equations, which fit neatly and naturally into one matrix of the tensor field. So you have four dimension coordinates of 10 functions with the gravity theory and five dimension coordinates of 15 functions with the Kaluza-Klein model.

"One cannot actually measure events in other than 3 dimensions; however, the time-distance coordinate is a way of describing an object's place in time as well as space, so the precise location of an object in spacetime requires 4 dimension coordinates -- x, y, z, t."

"Relativity, however, is a kinetic theory with roots in Einstein's early study of Brownian motion."

So we can only measure action in the three dimensional space. Wouldn't the most efficient solution be that the geometry arises from this action in space, not the motion in space arising from the geometry? How do you have kinetic action without the transfer of energy by which one event dissolves into a subsequent event, thus making only one event at a time physically possible?

The measurement is still of action. The question is whether this occurs in a physical dimension called 'time,' or if time is the result of the change caused by the action. Now obviously we are on opposite sides of this debate.

"What time interval are you referring to for 'one event at a time', and what do you mean by 'physically possible?''

Necessarily every action is creating the interval as a consequence of its process. For example, between peaks of waves; The duration between one and the next striking a sensor is due to a variety of physical factors; motion of the transmitter, frequency of the waves, any resistance, friction, other environmental factors. The point is that what is physically occurring is both the events and the processes leading from one to another. The present is not a point on some dimension of duration, but duration is what happens in the present between the occurrence of the events in question. It is this physically kinetic process of constant activity that creates the events which the geometry uses as points of measure. Spacetime is a map of the territory of events, not the physical basis for it.

The passage of time is a process, not an object. The verb, not another noun.

"The measurement is still of action. The question is whether this occurs in a physical dimension called 'time,' or if time is the result of the change caused by the action."

If that were the question, John, you still wouldn't have a way to measure time, and I would win the "debate" by default, because the action principle is implicit in a spacetime measure.

The real question is whether time exists at all. I'm with Minkowski and Einstein; only spacetime is physically real. You have yet to decide what is physically real -- you can use words like time, space, action, change -- until you quantify them, however, they mean whatever suits you in whatever argument you're trying to make. As we see. What is physically real? -- what do you think "physics" means?

What's this about 'quantify?' The system you insist is the word of god cannot explain why time is asymmetric, or why the present is what is physically real, so it insists it's all illusion. It cannot explain how gravity works, so we are looking for all the invisible patches, from inflation to dark matter to dark energy, that only the math sees. If you can afford to live in a world where only the math is real, than all those inconveniences can be ignored, but I have to live in one that makes sense, because there are too many unknowns to spend my time chasing ghosts.

Tom, I "root" for relativity. I think it's the Cinderella of contemporary physics. And IMHO part of the reason for that is the map is not the territory. Spacetime is the map. Space is the territory. Look up at the clear night sky. You don't see worldliness or lightcones. You see things moving. Ours is a world of space and motion. Spacetime models it, but is not it. Einstein gave us the equations of motion. Kinematics. And in spacetime there is no motion. Have a look at this: http://www.rebelscience.org/Crackpots/notorious.htm. Einstein and some others shouldn't be in the list. But the gist is correct. Spacetime is an all-times "block universe" mathematical model. It is totally static. A worldline represents motion through space over time. But there is no motion up that worldline. Next time you see a meteor, remember that.

Is there at all "a tangible effect on daily life"? To me, the blogger did not give an acceptable answer.

He asked "Infinity? Can it be a real thing rather than simply our idealization?"

To me infinity is simply an ideal logical concept, the property to be endless, which was formulated by Archimedes: There is no largest natural number. Cantor's naive set theory and its replacement by ZF intended to overcome this unwelcome property. In ZF, infinity is formulated as just one axiom among seven others including AC that were fabricated in order to castrate it for the sake of an illusive paradise.

If the notion infinity has a tangible influence then in so far as it's frequent mistreatment in set theory damaged the honest strive for consistency in science.

Leibniz who also introduced a not strictly endless relative (potential) infinity which is arbitrarily large or arbitrarily small (infinitesimal) understood what Cantor called the infinitum absolutum, as a fiction with a fundamentum in re.

Hence, outside established inconsistency there are logically consistent meanings of both Archimede's potential infinity and the fictitious limit alias infinitum absolutum which I often used in EE.

This is my answer to the blogger's question whether infinity can be a real thing rather than simply our idealization: Isn't our notion of reality an idealizing tool? We may not expect an answer, neither from current nor from a fundamentally corrected mathematics. Guth's speculation on t=0 will certainly not be useful in daily life.

Might the Infinity of the Universe Have a Tangible Effect on Daily Life?

I will answer Yes. But before some of these things can be discerned, there is a need to look at Clausius description of entropy, mathematically written as

dS = dE/T,

where S is entropy, E is energy and T is the absolute temperature, d stands for differential or change in value.

Taking all the other laws of thermodynamics as correct, what is the cosmological significance of this equation if the universe obeys it?

It is clear that the only way to obtain infinite entropy from this equation is for T to equal zero at the moment of energy change, dE. If the universe began from 'nothing' rather than a 'hot thing' what was the initial temperature at time zero? If the universe began from nothing, what was its initial entropy at time zero? Does the third law not say, when T = 0, S = 0 and vice-versa? What does energy change, dE mean? Can an isolated system change in energy without breaking a thermodynamic law? Can energy have a positive and a negative aspect so that the sum of the two in a conserved system still equals zero thereby preserving the thermodynamic law? More later...

Infinity negates entropy. Entropy only applies to closed sets loosing usable energy, yet in an infinite context, energy being radiated away is replaced by energy radiating in from an infinity of sources.

There is no beginning or end of the energy, because there is no way to cancel it. For all the positive and negative elements to cancel out, it would have to do so over an infinite space to stop the universe. Beginning and ending only really applies to form, which are constantly coming into being and dissolving.

"Which goes to the necessity of spatial and by extension, temporal infinity."

Careful, John, you're getting dangerously close to realizing why spacetime has to be a continuous physically real object.

Spacetime measures are self limited short of infinity by a sign change in the metric signature. Because there are no isolated systems, spacetime curvature which we measure as the partial boundary of a gravitational field influence, is fully bounded by all gravitatational influences (Mach's principle). This is why I claim that the equations of general relativity remain unchanged when the theory is inverted from Einstein's "finite and unbounded" description of a universe bounded in time (by a big bang event) and unbounded in space (by the return of a geodesic to its starting point) -- into a universe bounded in space (by a single point at infinity) and unbounded in time (by evolving relations between points of arbitrarily chosen initial condition).

Joy Christian's framework is fully compatible with this picture.

We really don't yet know what entropy means, in any sense that can be called complete. We do know, however, what it means to have a framework of complete measurement functions.

Is it possible there are other configurations to which these equations might equally apply, possibly even one similar to the reality we perceive, of an infinite, Euclidian space, with change and its various temporal measures, arising from the varying dynamic actions? If it is such a cycle of contracting mass and expanding radiation, with space treated as a measure of mass points and the cosmological constant to keep it stable, the geodesic would return to its mathematical starting point, as a revolution of the cycle and time is bounded by that which is present.

This way, neither space or time are physically real, but for different reasons. Space because it has no quantifiable physical properties to bend, bound, divide, energize, locate, energize, etc. Simply just the infinite void. Meanwhile time is simply a consequence of action, like temperature. That of the rate of change.

"We really don't yet know what entropy means, in any sense that can be called complete."

Isn't the assumption of 'complete' knowledge a presumption of finite knowledge?

Generally entropy would seem to refer to the thermal equilibrium to which a system evolves.

"There would therefore be no accelerated motion (rate change of rate of change) in your world.'

Why not? I frequently compare it to frequency and frequencies certainly vary. Music would be rather boring otherwise. The point is that it is a measure of the action and actions do change rates, much like accelerating a car.

"Where did I say anything about complete knowledge?"

"We really don't yet know what entropy means, in any sense that can be called complete."

"Equilibrium doesn't evolve."

I said the system evolves toward an equilibrium. Stephen Jay Gould's 'punctuated equilibrium' would be a good example of how ecological systems settle into increasingly stable patterns, between interruptions.

which is to say when a quantity of heat/energy is introduced into a system, part of it goes towards increasing entropy, dS and a part to do work, dW (= pdV + VdP). if we assume only a system with rigid borders can have pressure, then for a pressureless system and rearranging, we can say

which is to say when a quantity of heat/energy is introduced into a system, part of it goes towards increasing entropy, dS and a part to do work, dW (= pdV + VdP). if we assume only a system with rigid borders can have pressure, then for a pressureless system and rearranging, we can say

dE/T = dS.

If we add 100 joules of energy to a cup of tea in equilibrium at 10K temperature, the entropy will increase towards another equilibrium state, which will be by 10 J/K higher than the previous state.

The more the energy added, the more the disorder/entropy in the system.

Now, if a supreme being out of 'wickedness' or by design wants to increase the disorder, i.e. the entropy in a hitherto perfectly and infinitely ordered system, i.e. with S = 0 and he wants to do this such that system never again experiences order by making its subsequent equilibrium state to lie at a value at infinity, in what ways can this be done? One of the ways is by adding an infinite amount of energy, dE to an initial state at any temperature. This is expensive energy-wise. A wiser supreme being may be able to create infinite disorder in a much more economical way. That is all I am saying.

*I also observe that such a 'wicked' act will also cause a uni-directional entropy situation (a second law of thermodynamics). It will cause Time to come from nothing and 'flow continuously' since the system has to now continually change instead of remaining the same and evolve towards some infinite value of entropy. We will be under a curse to evolve towards thermal equilibrium but never getting there. *Since according to Boltzmann, entropy is related also to the number of constituents and the number of ways they can be arranged (S = klnW), the bigger the compartment and the more the number of constituents, the more the number of different possible arrangements and the disorder, it becomes compelling that for the value of entropy to be capable of being further increased, the number of constituents and the compartment size must after some limit be forced to increase to accommodate the 'curse' upon us in obedience to what the equation, dS = dE/T dictates.

"The point is that it is a measure of the action and actions do change rates, much like accelerating a car."

The car is accelerating in relation to what? What action is being measured?

"I said the system evolves toward an equilibrium. Stephen Jay Gould's 'punctuated equilibrium' would be a good example of how ecological systems settle into increasingly stable patterns, between interruptions."

Understand the mathematics that drives the model (self organized criticality) and you will see that systems do not become "increasingly stable."

"The more the energy added, the more the disorder/entropy in the system."

If the system is closed. We don't really understand what entropy means in general. Speaking of adding energy from outside the system doesn't really tell one what is going on inside -- one can only say that entropy doesn't decrease in a system left to itself.

"The car is accelerating in relation to what? What action is being measured?"

Is this a trick question?

The ground. Time as an effect and measurement existed long before General Relativity. The actions used to define and measure it were every bit as subjective as a moving vehicle. Swinging pendulums, celestial bodies, heartbeats, etc. While the cosmos tends not to speed up or slow down, heart rates certainly do. You're not explaining why treating time as a measure of action would mean the rate of change could not vary.

"you will see that systems do not become "increasingly stable." "

"entropy doesn't decrease in a system left to itself."

Then lets just say the entropy increases. Usually, say in a simple system of hot and cold fluids, some relatively stable state will be reached. Call it 'warm.'

"If the system is closed...". That is the most difficult problem for a cosmological model of a universe beginning from nothing. Edward Tryon proposed an energy fluctuation from nothing and tried to see how the energy conservation law could still remain intact (see links here and here. Whichever way it is, just thinking of it is a 'cool' headache: for something to arise from nothing! Eating your cake and not just having it but getting a bigger cake! OMG! Even in a religious context, a God would have emerged from nothing if there was a beginning to "everything" and to my thinking it appears simpler for a tiny amount of energy to arise out of nothing than for a giant omniscient and omnipotent being to do same.

But let's consider a system that is not closed. If you add energy to it, its entropy I think still increases? So if you have an infinitely small open system and introduce energy into it when it is at zero kelvin what would happen? By how much will its entropy increase? (I acknowledge the difficulty with absolute zero, so lets just say as close to zero as possible).

"His musings remind me of one of the strangest concepts in mathematics: the axiom of choice. This is a rule for selecting objects from an infinite collection even when all standard rules fail. The weird thing is that, although mathematicians know that such a rule exists, they don't know what the rule is. Worse, they know they'll never know what it is. A state of infinite entropy may likewise exist even if it is impossible to specify."

FWIW, in a 2006 conference paper I showed how to derive a well ordered continuum from random events, without appealing to the AC.

I know it's heretical to challenge the conservation laws, but seeing as how we don't really have a general definition of 'energy' which manifests itself in myriad forms, might it be an emergent phenomenon of spacetime? It is profoundly transient, and we can't truly say that the space-like dimension and the time-like dimension are themselves different things.

It is a long way from the laboratory of Lavoisier to intergalactic space. So does it matter if we see the universe as infinite or finite to accept covariance creating 'energy'? jrc

If I am right: first you put forwards the lack of a good understanding of the nature of energy (physics). Second you put a question mark by the importance of the existence of a finite/infinite universe in relation to the conservation of energy.

Well, I cannot say there is no good understanding, because energy creates differences in the properties of the phenomena and vice...

If I am right: first you put forwards the lack of a good understanding of the nature of energy (physics). Second you put a question mark by the importance of the existence of a finite/infinite universe in relation to the conservation of energy.

Well, I cannot say there is no good understanding, because energy creates differences in the properties of the phenomena and vice versa. With respect to the known phenomena we even know that the amount of energy of each phenomenon is a number of quanta. Nevertheless, the distinct properties of phenomena do not reflect nice proportions, therefore energy itself represents at least 2 mathematical quantities (otherwise the properties of phenomena were all integers).

Spacetime is a construction of Albert Einstein and the description is completely related to the “behaviour” of phenomena. Energy is more foundational so it is the opposite: spacetime (if it is real) is an emergent phenomenon of energy.

The importance of finite/infinite in relation to the law of energy conservation.

First, our opinion about the hierarchy in space has changed over the last century. The properties of phenomena are not properties that originate from the distinct phenomenon, but these properties reflect the local interactions between fields. So there are “primary” fields all over the universe (scalar and vector fields) and local deformations (local fields) that form the known phenomena (and undoubtedly a bit more we are not aware of).

Every field has a volume and a surface area (delimited in space). This volume and surface area represent an amount of energy. So the whole universe is a set of diverse energy volumes and the energy of these volumes interact with each other. But the total amount of energy volume has to be invariant (otherwise there were no physic laws). Now suppose the universe is not infinite. Will it be a problem for the energy conservation? Well, as long as the volume itself is invariant, all energy in the universe is conserved.

How do we notice the presence of the surface area of a local field? Because of the delimitation we are aware of the presence of the phenomena and the diverse differences between them. Now imagine our universe is limited. Therefore, the volume is limited and the surface area is...?

To limit the surface area of the universe you need to envelope the limited volume of the universe. But when 2 systems interact, there is an underlying foundation (set theory). So there is no envelope... Unfortunately, without an envelope the total amount of surface area is not conserved. And if that is true, we never had observed equal properties under equal circumstances. Therefore, we have to accept: our universe is infinite.

Thanks for your views, I like to see what different perspectives reveal.

If I understand you properly,('the total amount of energy volume must be invariant') the volume that a discrete quantity of energy would assume in reality would always be describable by a set of mathematic equations that would be the same for any quantity, in the sense that we hypothetically prescribe an idealized background independent theoretical rest mass. There-in; the energy would be distributed in a continuous range of density. jrc

"The aficionados of infinity include Alan Guth, who argued in Puerto Rico and on Edge.org that a truly infinite universe would neatly explain the arrow of time. When space has no bound, neither does entropy. It keeps on increasing forever, always pointing the way forward for time. The universe need not have begun in a contrived initial state to create the impetus toward increasing disorder."

I might have a comment to make with regard to the part "When space has no bound, neither does entropy. It keeps on increasing forever, always pointing the way forward for time." First I need to know if it is acceptable to say that the 'cosmological principle' includes a quasi-statically changing' temperature for the universe?

I am assuming that the type of entropy referred to is intended as thermodynamic entropy defined by Clausius. It is important to specify this because none of the other types are the same thing at all. The type of entropy was not stated. However, the thermodynamic 'arrow of time' originated with Clausius' definition of thermodynamic entropy.